We know that $$\int_{a}^{b}f(x)\:dx = -\int_{b}^{a}f(x)\:dx$$ However, when it comes to scalar line integrals, we know that $$\int_{C}g(x,y)\:dr=\int_{-C}g(x,y)\:dr$$ where $-C$ represents the parameterisation in the reverse direction.
Doesn't this create a paradox? Suppose I have two parameterisations $$p_1(t)=[t\;\;0]$$ $$p_2(t)=[(a+b-t)\;\;0]$$
where $a \leq t \leq b$
So $p_1(t)$ runs from $a$ to $b$ whereas $p_2(t)$ runs from $b$ to $a$. Hence, $$\int_{a}^{b}g(t,0)\:dt=\int_{a}^{b}g(a+b-t,0)\:dt$$
The integral on the LHS says that the function $g(t,0)$ (which is now a single variable function), is being integrated from $a$ to $b$. The integral on the RHS says that the function $g(t,0)$ is being integrated from $b$ to $a$. According to the first equation I have mentioned, shouldn't these two be of opposite signs?
I understand the mathematical derivation of why the path direction doesn't matter. However, I am having a gap in intuition while comparing to the single variable case.